Visualize a four-by-eight squared chocolate bar that is cut: diagonally in the middle, upward between the first and second column of the now top half, and selectively around a chocolate piece located in the first row and first column. Now, if that particular chocolate piece is removed from the whole bar; and if the top half is then slid in a downward position, while the first column replaces the last, then the four-by-eight squared chocolate is perceived to stay the same—as if the isolated chocolate piece had been created out of thin air. However, the illusion alters the height, and in effect the volume as well, in the chocolate bar, but the adjustment was so miniscule that it goes unnoticed unless done repeatedly. This physical process does…show more content… Visually, the amount of whole numbers are twice as much as the amount of even numbers. However, that is intuitively incorrect because even though the set of whole numbers are abundant compared to the even numbers, every even number can be matched with a whole number. This is because there is a one-to-one correspondence between the two sets that proves them to be equal in size. Because of this, mathematician's conclude that instances like: adding one to infinity, dividing infinity by two, or even 2,000,000, would still result in infinity. This can be better explained in the Hilbert’s Paradox of the Grand Hotel.
Shapes, more specifically: circles, are another example of infinity and Hilbert’s Paradox. Suppose a 2-Dimensional circle is drawn on a sheet of paper and a dot around the circumference of the circle is erased. This one point taken out of the circumference does not affect the circle, according to infinity, because the formula is irrational (C=2πr). If points are marked around the circle every “radius length” clockwise; and move these points counterclockwise like you would in an infinite hotel, then you obtain an endless supply of points that can cover the missing point (Vsauce,…show more content… It is orchestrated with objects that we have not been able to fully comprehend. Only 4.83% of our cosmos contains heavy elements, neutrinos, stars, and free hydrogen and helium. Those of which we cannot entirely understand, we also have dark matter and dark energy—a total of 95% (Templeton, 2015). As Yanofsky stated in The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us, “It is important to realize that ideas about infinity are not abstract scholastic thoughts that plague absentminded professors in the ivy-covered towers of academia. Rather, all of calculus is based on the modern notions of infinity mentioned in this chapter. Calculus, in turn, is the basis of all of the modern mathematics, physics, and engineering that make our advanced technological civilization possible. The reason the counterintuitive ideas of infinity are central to modern science is that they work. We cannot simply ignore them.” Infinite sets go against our common sense because it is uncommon, which is why it sounds counterintuitive. Banach-Tarski is not incorrect and has factual evidence to support it; but, because it is such a difficult and abstract idea to grasp, it is written off as pure conjecture. The theorem derives from common sense and enters a realm of the
